None of the options are correct. Let me give counter-examples:-
A) w = 'abbb'. M would accept but L won't.
B) w = 'b'. M won't accept but L would. This example is also applicable to option A and C.
C) w = 'aabb'. M won't accept but L would.
D) Same as option B.
Please don't say that 'a' has to be there in w. Then {a,b}* won't have any significance. Because the language produced by {a,b}* = {empty, a, b, aa, bb, aba, baa, ..............}.